structural damage evaluation: theory and

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STRUCTURAL DAMAGE EVALUATION: THEORY AND APPLICATIONS TO EARTHQUAKE ENGINEERING

Thesis by

Rustem Shaikhutdinov

In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

Report No. EERL 2004-06

California Institute of Technology Pasadena, California

2004

(Defended February 17, 2004)

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Acknowledgments

My sincere gratitude goes to my adviser Professor James L. Beck for his direction and encouragement throughout the course of this investigation. Without his invaluable guidance and support, this thesis would not have been possible. I am also truly grateful to Keith A. Porter for the suggestions and input that were most helpful. I am thankful for the delightful time we spent working together.

I truly appreciate the time and input of Professors Wilfred D. Iwan, John F. Hall in reviewing my thesis and delegating their time to serve on my thesis examination committee. It has been a really enriching experience to be their student, as well.

My studies were financially supported by funds from California Institute of Technology; California Universities for Research in Earthquake Engineering under the CUREe-Kajima joint research program, Phase IV; the Earthquake Engineering Research Centers Program of the National Science Foundation under Award Number EEC- 9701568 through the Pacific Earthquake Engineering Research Center (PEER). These supports are gratefully acknowledged. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the support providers.

I am thankful to Carolina Oseguera and Connie Yehle for always being so thoughtful and supportive throughout all the years of my staying at Caltech.

My special thanks go out to my lovely wife Julia, who gave me strength and determination to pursue on the road of research and instilled my days with joy and happiness.

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Abstract

The further development of performance-based earthquake engineering (PBEE) is on the current agenda of the earthquake engineering community. A part of assessing the seismic performance of civil engineering structures involves estimation of seismic damage. The conventional approach to damage estimation is based on fragility functions that relate some chosen parameters of structural response to incurred damage. Therefore, damage prediction is based exclusively on the knowledge of the chosen structural response parameters, meaning that damage analysis is uncoupled from the structural analysis. The structural response parameters selected for use in damage analysis are usually referred to as engineering demand parameters (EDP). In the present study, it is shown that for structural damage estimation, the uncoupled damage analysis has deficiencies that lead to less accurate damage prediction. These shortcomings originate from two sources: first, dependence of practically all EDPs on structural damage and second, inexact damage description. To overcome these deficiencies, another approach to structural damage estimation is proposed. The proposed approach, besides using an EDP, uses all information available from structural analysis that is relevant to the damage to be assessed, implying that damage analysis is coupled with structural analysis. It is shown that utilization of this additional information provides more accurate damage prediction.

The difference between the two approaches is studied by comparison of results of damage estimation performed for a 2-D structural model of a reinforced-concrete frame.

The results show that difference between uncoupled and coupled damage analysis estimates could be significant and that it depends on specific characteristics of the chosen structural model and the damage model in a complex way, preventing the possibility of

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estimating this error in a general form that is applicable to all practically possible cases.

Damage estimation is performed for various damage models that include both single and multiple damage states. Since the final goal of seismic performance evaluation is estimation of decision variables such as repair cost, downtime, etc., the two approaches to damage estimation are also compared in terms of repair cost that is calculated for the reinforced-concrete frame. A case where structural damage prediction is based on observation of EDP alone, without a structural model available, is also studied. It is shown that incorporating site-specific information can significantly change the damage estimates and, therefore, may be worth doing.

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Table of Figures

Figure 2.1 Example of utility function for decision making based on safety. 7 Figure 2.2 Example of a decision making process for a real estate owner. 10 Figure 2.3 Implementation structure of the PEER PBEE framing equation. 22 Figure 3.1 Fragility functions for multiple damage states 47 Figure 3.2 Damage states probability space for different values of EDP 47 Figure 3.3 Original state space and redefined state space with corresponding probability

measures 51 Figure 4.1 Relations between the variables in the state space. 53

Figure 4.2 Uncoupled structural and damage analyses. 54

Figure 4.3 Coupled structural and damage analyses 55

Figure 4.4 Methods used for the sample case study damage estimation. 57 Figure 4.5. Reinforced concrete moment-resisting frame chosen for the case study. 59 Figure 4.6. Flexural members hysteretic rule: Q-HYST. 60 Figure 4.7 Relation between force-displacement history and observed damage states of a reinforced concrete column (Tanaka and Park, 1990) 65 Figure 4.8. Expectation of damage E[Nt] as a function of spectral acceleration, ground

motion LA15. 73

Figure 4.9. Relative errors of estimation of E[Nt] by different methods as a function of

spectral acceleration, ground motion LA15. 73

Figure 4.10. Variance of the damage estimate as a function of spectral acceleration,

ground motion LA15. 74

Figure 4.11. Coefficient of variation of the damage estimate as a function of spectral

acceleration, ground motion LA15. 74

Figure 4.12. Expectation of damage E[Nt] as a function of spectral acceleration, set of

ground motion records. 76

Figure 4.13. Relative errors of estimation of E[Nt] by different methods as a function of spectral acceleration, set of ground motion records. 77 Figure 4.14. Variance of damage estimation as a function of spectral acceleration, set of

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Figure 4.15. Coefficient of variation of the damage estimate as a function of spectral

acceleration, set of ground motion records. 78

Figure 4.16 Example of the yield surface of the flexural members (axial force - moment

interaction). 83 Figure 4.17. Expectation of damage E[Nt] as a function of spectral acceleration for the

axial force-flexure interaction model, ground motion LA15. 92 Figure 4.18. Relative errors of estimation of E[Nt] by different methods as a function of spectral acceleration for the axial force-flexure interaction model, ground motion

LA15. 92 Figure 4.19. Variance of damage estimation as a function of spectral acceleration for the

axial force-flexure interaction model, ground motion LA15. 93 Figure 4.20. Coefficient of variation of the damage estimate as a function of spectral

acceleration for the axial force-flexure interaction model, ground motion LA15. 93 Figure 4.21. Expectation of damage E[Nt] as a function of spectral acceleration for the axial force-flexure interaction model, set of ground motion records. 94 Figure 4.22. Relative errors of estimation of E[Nt] by different methods as a function of spectral acceleration for the axial force-flexure interaction model, set of ground

motion records. 94

Figure 4.23. Variance of damage estimation as a function of spectral acceleration for the axial force-flexure interaction model, set of ground motion records. 95 Figure 4.24. Coefficient of variation of the damage estimate as a function of spectral

acceleration for the axial force-flexure interaction model, set of ground motion

records. 95 Figure 4.25 Safety regions for exact (hatched) and approximate (shaded) limit-state

functions, fragility function is CDF of capacity at zero axial force 99 Figure 4.26 Using flexural member's yield surface for determining safety region in terms

of yield moment at zero axial force 100

Figure 4.27 Comparison of the safety regions defined by exact and approximate limit- state functions for the assumed probability distribution of current axial force. 103 Figure 4.28 Safety regions for exact (hatched) and approximate (shaded) limit-state

functions, fragility function is CDF of capacity at expected axial force 104

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Figure 4.29 Safety regions for exact and approximate limit-state functions, lower values of axial force expected value and standard deviation are used 105 Figure 4.30 Safety regions for exact (hatched) limit-state function and two-dimensional

approximate (shaded) limit-state function, axial force is relatively small 108 Figure 4.31 Procedure of finding safety region for two dimensional fragility function,

axial force is relatively high 108

Figure 4.32 Safety regions for exact (hatched) limit-state function and two-dimensional approximate (shaded) limit-state function, axial force relatively large 109 Figure 5.1 Hysteretic rule for flexural members (Q-HYST with strength degradation). 112 Figure 5.2 Expected number of flexural members in three damage states of interest,

obtained by Method 1, correlation of capacities 0.6, ground motion LA15. 120 Figure 5.3 Expected number of flexural members in three damage states of interest,

obtained by Method 2, correlation of capacities 0.6, ground motion LA15. 121 Figure 5.4 Expected number of flexural members in three damage states of interest,

obtained by Method 3, correlation of capacities 0.6, ground motion LA15. 121 Figure 5.5 Expected number of flexural members in DS = 1 for multiple damage states model with correlation of capacities 0.6, ground motion LA15. 122 Figure 5.6 Expected number of flexural members in DS = 2 for multiple damage states

model with correlation of capacities 0.6, ground motion LA15. 122 Figure 5.7 Expected number of flexural members in DS = 3 for multiple damage states

model with correlation of capacities 0.6, ground motion LA15. 123 Figure 5.8 Variance of number of flexural members in DS = 1 for multiple damage states model with correlation of capacities 0.6, ground motion LA15. 123 Figure 5.9 Variance of number of flexural members in DS = 2 for multiple damage states model with correlation of capacities 0.6, ground motion LA15. 124 Figure 5.10 Variance of number of flexural members in DS = 3 for multiple damage

states model with correlation of capacities 0.6, ground motion LA15. 124 Figure 5.11 Coefficient of variation of number of flexural members in DS = 1 for

multiple damage states model with correlation of capacities 0.6, ground motion

LA15. 125

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Figure 5.12 Coefficient of variation of number of flexural members in DS = 2 for multiple damage states model with correlation of capacities 0.6, ground motion

LA15. 125 Figure 5.13 Coefficient of variation of number of flexural members in DS = 3 for

multiple damage states model with correlation of capacities 0.6, ground motion

LA15. 126 Figure 5.14 Expected repair cost estimate for multiple damage states model with

correlation of capacities 0.6, ground motion LA15. 126 Figure 5.15 Variance of repair cost estimate for multiple damage states model with

correlation of capacities 0.6, ground motion LA15. 127 Figure 5.16 Coefficient of variation of repair cost estimate for multiple damage states

model with correlation of capacities 0.6, ground motion LA15. 127 Figure 5.17 Expected number of flexural members in three damage states of interest,

obtained by Method 1, correlation of capacities 0.6, set of ground motion records.

128 Figure 5.18 Expected number of flexural members in three damage states of interest,

128 Figure 5.19 Expected number of flexural members in three damage states of interest,

129 Figure 5.20 Expected number of flexural members in DS = 1 for multiple damage states model with correlation of capacities 0.6, set of ground motion records. 129 Figure 5.21 Expected number of flexural members in DS = 2 for multiple damage states model with correlation of capacities 0.6, set of ground motion records. 130 Figure 5.22 Expected number of flexural members in DS = 3 for multiple damage states model with correlation of capacities 0.6, set of ground motion records. 130 Figure 5.23 Variance of number of flexural members in DS = 1 for multiple damage

states model with correlation of capacities 0.6, set of ground motion records. 131 Figure 5.24 Variance of number of flexural members in DS = 2 for multiple damage

states model with correlation of capacities 0.6, set of ground motion records. 131

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Figure 5.25 Variance of number of flexural members in DS = 3 for multiple damage states model with correlation of capacities 0.6, set of ground motion records. 132 Figure 5.26 Coefficient of variation of number of flexural members in DS = 1 for

multiple damage states model with correlation of capacities 0.6, set of ground

motion records. 132

Figure 5.27 Coefficient of variation of number of flexural members in DS = 2 for multiple damage states model with correlation of capacities 0.6, set of ground

motion records. 133

Figure 5.28 Coefficient of variation of number of flexural members in DS = 3 for multiple damage states model with correlation of capacities 0.6, set of ground

motion records. 133

Figure 5.29 Expected repair cost estimate for multiple damage states model with correlation of capacities 0.6, set of ground motion records. 134 Figure 5.30 Variance of repair cost estimate for multiple damage states model with

correlation of capacities 0.6, set of ground motion records. 134 Figure 5.31 Coefficient of variation of repair cost estimate for multiple damage states

model with correlation of capacities 0.6, set of ground motion records. 135 Figure 5.32 Expected number of flexural members in DS = 1 obtained by Method 1 for low (0.6) and high (0.9) coefficient of correlation of member capacities, ground

motion LA15. 135

Figure 5.33 Expected number of flexural members in DS = 2 obtained by Method 1 for low (0.6) and high (0.9) coefficient of correlation of member capacities, ground

motion LA15. 136

motion LA15. 137

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motion LA15. 137

motion LA15. 138

Figure 5.38 Variance of number of flexural members in DS = 1 obtained by Method 1 for low (0.6) and high (0.9) coefficient of correlation of member capacities, ground

motion LA15. 138

motion LA15. 139

motion LA15. 140

motion LA15. 141

Figure 5.44 Expected repair cost obtained by Method 1 for low (0.6) and high (0.9) coefficient of correlation of member capacities, ground motion LA15. 141 Figure 5.45 Expected repair cost obtained by Method 2 for low (0.6) and high (0.9)

coefficient of correlation of member capacities, ground motion LA15. 142 Figure 5.46 Variance of repair cost obtained by Method 1 for low (0.6) and high (0.9)

coefficient of correlation of member capacities, ground motion LA15. 142

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Figure 5.47 Variance of repair cost obtained by Method 2 for low (0.6) and high (0.9) coefficient of correlation of member capacities, ground motion LA15. 143 Figure 5.48 Expected number of flexural members in DS = 1 obtained by Method 1 for low (0.6) and high (0.9) coefficient of correlation of member capacities, set of

Figure 5.49 Expected number of flexural members in DS = 2 obtained by Method 1 for low (0.6) and high (0.9) coefficient of correlation of member capacities, set of

Figure 5.54 Variance of number of flexural members in DS = 1 obtained by Method 1 for low (0.6) and high (0.9) coefficient of correlation of member capacities, set of

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Figure 5.60 Expected repair cost obtained by Method 1 for low (0.6) and high (0.9) coefficient of correlation of member capacities, set of ground motion records. 149 Figure 5.61 Expected repair cost obtained by Method 2 for low (0.6) and high (0.9)

coefficient of correlation of member capacities, set of ground motion records. 150 Figure 5.62 Variance of repair cost obtained by Method 1 for low (0.6) and high (0.9)

coefficient of correlation of member capacities, set of ground motion records. 150 Figure 5.63 Variance of repair cost obtained by Method 2 for low (0.6) and high (0.9)

coefficient of correlation of member capacities, set of ground motion records. 151 Figure 6.1 Repair cost estimates based on combined method of damage analysis, set of

Figure 6.2 Expectation of repair cost estimate and one-sigma confidence intervals obtained by combined method of damage analysis, set of ground motion records. 157 Figure 6.3 Standard deviation of repair cost estimate obtained by combined method of

damage analysis, set of ground motion records. 157

Figure 7.1 Column model 161

Figure 7.2. Force-deformation characteristics of the shear spring (left) and flexural

spring (right). 162

Figure 7.3 Failure region and safe region in the Ω space. 164 Figure 7.4 Surface Ωz for different values of IDR 166

Figure 7.5 Surface Ωz for different values of α 166

Figure 7.6 Site-specific fragility function and CDF of capacity for the "normal strength"

design. 171

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Figure 7.7 Site-specific fragility functions for “normal strength” design for different

values of α 172 Figure 7.8 Site-specific fragility function and CDF of capacity for the "over-strength"

design. 174 Figure 7.9 Site-specific fragility functions for “over-strength” design for different values

of α 174 Figure 7.10 Site-specific fragility function and CDF of capacity for the "under-strength"

design. 175 Figure 7.11 Site-specific fragility functions for “over-strength” design for different values

of α 175 Figure 7.12 Site-specific fragility functions for the "normal strength", "under-strength"

and "over-strength" design ( α= 0.0001). 176

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1 Introduction

The primary focus of the present study is structural damage estimation. Damage estimation is a vital part of the seismic performance evaluation of buildings and other structures with respect to multiple performance objectives. In turn, the proper evaluation of seismic performance is essential for decision making involved in managing the risk to building, bridges, and other infrastructure in seismically active areas. Today, the earthquake engineering community faces new challenges that are brought about by the latest needs of the real estate development and management industries. The safety of buildings and other structures used to be the main concern of designers, owners, and regulators. The development of modern building codes has provided society with guidelines that serve well for achieving the required safety levels. However, nowadays other issues are becoming significant for owners and risk managers. Providing that safety requirements are met, the questions being asked now are “how much does it cost to repair?”, “how long it will be shut down in case of the earthquake?”, etc. These questions relate to the economic aspect of the seismic performance of real estate. Given the multiple performance objectives, accurate damage estimation becomes more important than ever. The issues involved in the decision-making process with respect to various performance objectives and the role of damage estimation in this process are discussed in detail in Sections 2.1 and 2.2, respectively.

Depending on the decision maker’s needs, a building’s seismic performance can be evaluated on different levels of accuracy. A rough performance estimate can be obtained by category-based techniques, where all facilities are subdivided into classes

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that are specified by some formal feature and/or parameters. For example, such classes may be: reinforced-concrete shear-wall building higher than 7 stories, or single-span bridge with monolithic abutment. Then a performance prediction model is built on the observed seismic performance of all available samples within the category. This approach is used, among others, by Hart and Srinivasan (1994) and by Basoz and Kiremidjian (1999).

More advanced techniques include specific building information in the damage analysis, such as particular design features and the site seismic hazard. The information about structural design is usually included in a finite element structural model. The structural model is used for carrying out a structural analysis. Damage analysis is then executed based on the results of the structural analysis. If better performance prediction is desired by the decision maker, then more site-specific information can be incorporated in the structural analysis and the analysis can be closer to the real-life behavior of the structure. For a relatively rough analysis, a simplified building model, like a shear beam, can be used in a pushover analysis. If more accurate results are desired, then more sophisticated structural models should be used, such as detailed finite-element models together with dynamic time-history simulation. Accordingly, the damage estimation technique should match the accuracy of the structural analysis.

Here is where the key point of the present study lies. It is apparent that structural analysis tools are making steady progress in increasing the accuracy of models. More and more features of real structure behavior are included in the mathematical models, such as shear-flexure interaction or complex load interaction in 3-D. However, it will be shown that current damage analysis techniques in some cases do not match the enhanced

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performance of the advanced structural analysis tools, making these advances largely wasted for loss estimation studies. Indeed, since the seismic performance assessment is based both on structural analysis and damage analysis, the inaccurate damage evaluation prevents reliable performance estimation from being achieved. The highly accurate structural analysis will not contribute to the final goal – accurate performance estimation, because the structural analysis results will be diluted by the larger errors at the stage of the damage estimation. Thus, the goal here is to develop improved methods of damage estimation that are capable of providing the results with an accuracy that matches the accuracy of the most sophisticated structural models.

From a practical perspective, as it is mentioned before, if more precise performance estimation is desired, then more site-specific information should be included in the analysis and more complicated analysis should be performed. It is true for the structural analysis and it is true for the damage analysis. In general, this means that the better performance estimation requires more effort and expense to perform the analysis.

Such efforts are not justified for all real estate risk management problems. In most cases a high-end solution is unnecessary. However, there are problems when maximum accuracy is necessary despite the cost of achieving it. This is the case when calibration of less complex methods of performance analysis is needed; that is, the sophisticated method must provide the most accurate solution so that the performance of simplified techniques may be judged by how close their results are to the “exact” one. This is also the case when evaluation of the unique and expensive facilities is performed, where the additional expense for the more complicated analysis is justified.

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Before we proceed with the introduction to the different damage estimation approaches, we want to point out that increasing utilization of site-specific information in the seismic performance analysis can lead to a potential problem, related to the inherent difficulty of experimental verification of the site-specific analysis results. A more detailed discussion of this problem is given in Section 2.3. Another problem of seismic performance evaluation that can arise is due to the different sensitivity of the decision- making process to the errors in performance assessment for different decision criteria.

The issues relevant to this problem are also considered in Section 2.3

At the present time, the common tools of general purpose damage analysis are fragility functions that are used for uncoupled damage analysis. These functions establish a probabilistic relation between structural response to seismic loading and the resulting damage to individual components. Historically, fragility functions were developed within nuclear engineering to evaluate the seismic resistance of nuclear reactors with respect to operational or safety failure. Later, their area of application was extended to other fields, such as estimation of the seismic resistance of electrical equipment or the seismic performance of civil engineering structures such as bridges and buildings. For all these applications, fragility functions proved to be convenient, versatile and reasonably accurate tools of damage estimation.

In order to use fragility functions for a general, multiple criteria performance analysis, they must be applied to a broader range of objects than previously and serve for evaluation of a broader range of issues. The objects for damage evaluation can be any damageable piece of the structure or nonstructural components, such as beams, columns, partitions, windows, equipment, furniture, etc. The loading parameters (input to a

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fragility function) can be spectral acceleration Sa, inter-story drift ratio (IDR), ductility demand, peak ground acceleration (PGA), etc., with the latter having been most heavily utilized before. The assessed damage can be fracture, full or partial loss of functionality, toppling, leakage, cracking, spalling, etc. As we can see, this amounts to a significant extension of fragility functions’ applications. The adequacy of fragility functions and uncoupled damage analysis as tools to handle all of this newly introduced variety of problems can not be taken for granted. Indeed, we shall see that fragility functions as a part of uncoupled damage analysis have certain limitations, especially in the area of structural damage estimation. Chapter 3 provides a rigorous mathematical description of fragility functions from the perspective of structural reliability theory. The theory identifies the limitations of fragility functions and provides a foundation for devising a coupled approach to structural response and damage estimation.

Using the results presented in Chapter 3, we have developed a coupled approach to structural analysis and damage estimation that does not have the shortcomings of the uncoupled damage analysis. Within the proposed approach, we have developed a method of damage estimation that is referred to as Method 1 henceforth. Seismic damage has been estimated for some chosen case-study facilities by the proposed method and also by the uncoupled method. Chapter 4 describes the case studies and provides the results of damage estimation performed by the different techniques.

The problem with multiple damage states is explored in Section 3.2 and Chapters 5 and 6. Section 3.2 discusses theoretical issues arising from the multiple damage state case. Chapter 5 presents a case study of structures that consist of members with multiple damage states. Damage of the structure is evaluated by the existing methods and by the

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proposed method of damage analysis. The same chapter describes how repair cost can be calculated based on the known damage state. The repair cost estimates are obtained for damage states that have been evaluated by different damage analysis methods. Results are compared in terms of both a damage measure (number of damaged members) and repair cost. A combined approach to damage analysis is developed in Chapter 6. To demonstrate a practical value of the proposed combined damage evaluation technique, the repair cost of an example structure is evaluated.

Chapter 7 deals with the problem of using fragility functions for structural health monitoring applications. It considers a model of a reinforced-concrete column and demonstrates that fragility functions applicable to in-situ structural members may differ from fragility functions that are developed using experimental data from in-lab test results.

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2 Methods and challenges of performance-based earthquake engineering

2.1 Decision making for a real estate owner

Seismic performance is a vital characteristic of buildings and other structures for all agents that are involved in operations with real estate located in seismically affected areas. How well a particular building will perform during an earthquake at some point in the future is important because it affects the present value of the property. In particular, at any present time, a real estate owner can face a set of seismic risk-management options to choose from: do nothing, sell the property, perform seismic retrofit or buy earthquake insurance. Likewise, a potential owner (a person who wants to buy a real estate property) faces similar choices: do not buy, buy and do nothing, buy and retrofit, buy and insure.

Figure 2.1 Example of utility function for decision making based on safety.

The process of making a choice between several alternatives can be analyzed by decision theory. Here we outline a simple procedure of formal decision making process.

This analysis does not consider uncertainty in the outcomes or risk preferences of decision makers. The general approach of decision theory states that the best choice is the one that gives the highest utility among different options (for details about utility and

Safety Utility

Sac

U(Sac)

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decision theory see, for example, Resnik 1987). Calculation of utilities for different options depends on the decision maker’s objectives and preferences. When applying this concept to the case of a real estate owner or a buyer, usually the most prevalent concern is safety. In terms of decision theory, this means that the higher the safety of some option, the higher is its utility, meaning that utility is the increasing function of safety. Normally, it suffices to use a very simplistic utility function to account for the matter of safety. It is convenient to utilize a step function like one shown in the Figure 2.1. Such function basically states that any option with the safety less than some acceptable level should be rejected. When the safety is higher than Sac, the utility is constant, implying that there is no marginal benefit from increasing safety beyond the acceptable level. This situation reflects an approach of real estate owners, where Sac represents the safety level provided by modern building codes. Alternatively, for some owners, the acceptable level of safety is the one that meets minimum legal requirements. In both cases, once the safety requirement is satisfied, he or she does not care if the safety level is significantly higher than Sac or just barely exceeds the threshold value.

If several options meet the safety requirement, then the owner can use some other criteria in order to choose between these options. For many owners, it appears that the other important issue that affects decision making is economic performance. There are different ways one can measure the economic performance of buildings and other structures with respect to seismic activity. Some of the quantities that are important for economic performance are repair cost and downtime. The repair cost represents the direct losses that the owner can incur – it is an uncertain expense associated with owning real estate. Downtime is the total time required to make the facility operational again and it

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can be related to both direct and indirect losses, where “direct” losses are understood as decrease of value (e.g., cash outflow) and “indirect” losses refer to missed opportunities to acquire value (loss of profit due to business shut down). For example, for the residential owner, the downtime could be the time when the building is not livable.

Therefore, temporary lodging would be needed, which means that the owner would bear the direct rental expenses associated with the lodging. For the commercial real estate, the downtime induces a loss of profit, which may be considered as the indirect loss. Clearly, both direct and indirect potential seismic losses associated with different options can affect the owner’s choice. Depending on the preferences of a particular owner, a utility function can be defined for repair cost and downtime.

Implementation of the decision choices usually involves some cost for the decision maker. For example, if a potential buyer decides to buy a property he will have to pay the purchase price. Similarly, if he resolves in favor of retrofitting measures or purchasing earthquake insurance, he will have to bear the price of the retrofit or insurance premium. The cost of each choice has a certain negative utility for the decision maker as well. Summing the utilities of the implementation cost and the future benefits, the option with the highest utility can be found. An illustration of the decision-making procedure for a current building owner is shown in Figure 2.2. In this example, the building owner can choose between three options: retrofit, insure or do nothing. By analysis, the owner can determine the option that maximizes his utility. In this case, all choices are associated with losing value: retrofit cost, insurance premium, future deductible and future losses that can include repair cost and downtime. Therefore, the best option minimizes the total loss.

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Figure 2.2 Example of a decision making process for a real estate owner.

The decision making process presented by Figure 2.2 disregards the uncertainty of the future losses and discounting of the future losses. A more sophisticated analysis would use the expected value of the utility of the future losses and benefits discounted to the present time, that is, use their present value (PV); see, for example, Beck et al. 2002.

For the purpose of the present study it suffices to note that the optimal choice is based on a formal decision making process. An essential part of this process is evaluation of future losses that can be expressed in terms of repair cost or downtime or some other parameters. In the following chapters, we shall consider the ways to calculate these parameters.

2.2 Methods of performance-based earthquake engineering

The decision-making procedure described in Chapter 2.1 can be performed only if reliable estimates of building performance are available. Obtaining such estimates is not a trivial problem. The quantities of interest, such as the expected number of lives lost, repair cost and downtime depend on a huge number of uncertain variables. The number

1. Retrofit U1 = U( Retrofit Cost + Future Losses )

2. Insure U2 = U(Insurance Premium + Future Deductible)

3. Do nothing U3 = U(Future Losses)

Ub = max(Ui)

Chose option with utility Ub

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of lives lost is one of the measures of the safety performance of buildings. The analysis of the safety of buildings and other structures has long been the subject of study of structural reliability theory. We give here a brief overview of the structural reliability approach.

2.2.1 Overview of structural reliability theory for seismic safety

Structural reliability theory for analysis of seismic safety usually does not directly consider such safety measures as expected number of lost lives. Instead, it deals with the events that can be directly related to the deaths caused by an earthquake. Such events are usually referred to as life safety failure (LSF). Two examples of LSF are total structural collapse and partial structural collapse. The problem of interest for practical applications is finding the probability of LSF. In general, this probability can be calculated according to the following probability integral

(

^LSF

)

^f_Q_X

( )

^q ^x^S ^d^q^d^x^S

P

F

∫

S

Ω

= _, , (2.1)

where Q is a vector of random variables that fully define the seismic excitation (ground acceleration time history is commonly used); X^S is a vector of random variables defining the values of all relevant structural properties; q and x^S are particular values of the random vectors Q and X^S, respectively; ^f_Q_,_X^S

( )

^q^,^x^S is the joint probability density function of random vectors Q and X^S; and ΩF is the failure region comprising all the values of Q and X^S for which LSF occurs.

For convenience of calculation, the failure region is usually given a mathematical description as follows. Define a function ^g

( )

^q^,^x^S in such a way that it possesses the following property

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( )

^q^,^x^S ^<⁰

g Ù [q,x^S]∈Ω_F (2.2)

meaning that region of negative values of ^g

( )

^q^,^x^S coincides with the failure region.

Function ^g

( )

^q^,^x^S is called a limit-state function for the LSF. Then the probability of LSF becomes

( ) ( )

( )

S x

q g

S X

Q q x dqdx

f LSF

P

S

∫

S

<

=

0 ,

, , (2.3)

Limit-state functions can be defined in a number of ways. One example is to define it in terms of maximum inter-story drift ratio (IDR)

( )

^q ^x^S

g , = dl −dm

( )

q,x^S (2.4) where dm is the maximum IDR resulting from a particular earthquake excitation q applied to a structure with properties x^S; dl is a chosen threshold value. This limit-state function implies that life safety failure occurs once the threshold value is exceeded: dm >

dl. Therefore, this approach assumes that it is likely that the structure undergoes partial or complete collapse once the maximum IDR exceeds the threshold value. The choice of threshold value depends on a structure type and may be based on experimental or field observations. Substituting (2.4) into (2.3), we obtain a special case of the structural reliability integral

( ) ( )

( )

S x

q d d

S X

Q q x dqdx

f LSF

P

S m l

∫

S

<

=

,

, , (2.5) Evaluation of integral 2.3 (or 2.5) is not a trivial task because vectors Q and X^S can contain up to several thousand variables and calculation of the function ^g

( )

^q^,^x^S ^is

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often computationally expensive because it involves a nonlinear structural analysis. A number of methods have been developed to estimate integral (2.3). Some of them are FORM, SORM and various Monte Carlo simulation based techniques (see, for example, Au and Beck, 2001a and Au and Beck, 2001b).

The well-developed methods of structural reliability are designed for estimation of the safety performance of structures. They do not provide ready tools for estimation of the economic performance of real estate. The development of such tools is on the current agenda of the earthquake engineering community. In this work, we shall discuss the approaches and methods used for development of such tools in detail.

2.2.2 Performance-based earthquake engineering framing equation

2.2.2.1 Standard integral form for arbitrary decision variables

Economic performance of real estate depends on more variables than safety performance. From Equation 2.1, it can be seen that in dealing with safety issues, structural reliability theory considers the sets of random variables that describe the ground motion and the structural properties. However, the knowledge of these variables is not sufficient for estimation of many important performance criteria, such as repair cost or downtime. For example, parameters like equipment price or labor cost are essential for estimation of the building repair/replacement cost. Including all the variables that affect the economic performance of the real estate into (2.1) leads to

P(EF) = ^f

(

^q ^x ^m

)

^d^q^d^x^d^m

EF

M X

∫

Q Ω

,

, ,

, (2.6)

where Q defines ground motion time history as before, X contains both structural and nonstructural properties of the real estate (nonstructural properties can be the properties

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of the building nonstructural elements or the building contents), M is composed of variables that represents market conditions (prices, availability of materials and contractors, etc.); ^fQ_,X_,M

(

^q,^x,^m

)

is a joint probability density function of all the variables; EF is an event that is classified as an economic failure and ΩEF is the failure region for EF. The term “economic failure” is chosen to be consistent with “life safety failure.” For different decision makers, EF can mean different things. For example, for an owner of commercial real estate, economic failure can mean that the repair cost is higher than some acceptable level. Alternatively, the owner may be intolerant to a downtime that is too long.

The limit function for EF is defined in a way similar to (2.2), with nonstructural properties and market conditions included in the independent variables

(

^q^,^x^,^m

)

^<⁰

g Ù [q,x,m]∈Ω_EF (2.7)

Depending on the priorities of the decision maker, the limit-state function can be defined in various ways. For example, it can be expressed in terms of repair cost or some other performance criteria. For repair cost, the limit-state function can be defined similarly to (2.4) as:^g

(

^q,^x,^m

)

=^Cl −^CR

(

^q,^x,^m

)

. For the present study, we shall use a general formulation of the limit-state function

(

^q ^x ^m

)

^DV ^DV

(

^q ^x ^m

)

g , , = _l − , , (2.8)

where DV (decision variable) stands for repair cost, downtime or any other variable that is important for a decision maker and DVl is a chosen threshold value. The name

“decision variable” is adopted because it is involved in the performance criterion that is

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used in the decision making process subsequent to the building evaluation. Using (2.7) and (2.8), we can rewrite (2.6) in the following way

P(DV > DVl) =

( )

( ) f q x m dqdxdm

DVl

m x q DV

M X

∫

Q

>

, ,

,

, , , (2.9)

Comparing (2.9) and (2.5), it can be seen that in general the problem of economic performance evaluation is more complex than the problem of safety performance evaluation. First, the number of integration variables is larger, implying a corresponding increase in the dimensions of the integration space and the computational effort. The second hurdle stems from the fact that the limit function (2.8) is more difficult to evaluate for economic decision variables than for decision variables that are typically used in safety performance analysis. We shall now discuss this problem in more detail.

2.2.2.2 Evaluation of damage and decision variables

Consider the limit function (2.4) for the safety integral (2.5). Besides the designated threshold value of IDR dl, it contains the maximum IDR which is a function of the ground motion and structural properties (d_m(q,x^S)). The problem of evaluation of

) ,

( ^S

m q x

d is equivalent to the problem of evaluation of the structural response of the building with the properties defined by x^S subjected to the seismic excitation q. This problem has been extensively addressed in the past. A number of structural simulation software packages are freely or commercially available. New tools (e.g., OpenSees, PEERC, 2004) are under development to provide better accuracy in structural simulation.

Therefore, the limit-state function (2.4) can be readily evaluated with existing tools.

Consider now the limit function (2.8) with the repair cost as DV. The repair cost of the whole structure can be found as a function of the repair or replacement cost of each

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of its damaged components. Therefore, in order to evaluate (2.8), we need to know the repair or replacement cost of each building assembly as a function of vector [q,x,m], where small letters, as before, stand for particular values of the corresponding random variables [Q,X,M]. For instance, take the repair/replacement cost of a window pane.

The question to answer is: what is this cost for the particular window given that the building is subjected to the ground motion q, properties of the building structural and non-structural components are x, and market prices for parts and labor are m? Let us assume that the window is not repairable, meaning that it can only be left as it is (if intact) or it can be replaced (if broken). Then if the market conditions are known, we know the cost of replacement of the window with the known dimensions and quality.

Also, given the ground motion q and building properties x, we can utilize structural properties x^S to obtain the structural response of the building (standard structural simulation packages can be used). With this kind of information, it is still unknown what has happened to the window. The window may or may not need replacement, meaning that the window replacement cost may be the market price of the replacement or zero.

Therefore, we can not determine the exact value of the window repair/replacement cost and, consequently, we can not evaluate the limit-state function (2.8).

To find the window repair/replacement cost, we need additional information. This additional information is associated with the damage of the window. It can be easily seen that once we know how badly the window is damaged, we can determine the consequences with a high degree of certainty. For the window example, the repair/replacement cost can be defined with absolute certainty if the market conditions are also known, where the known market conditions mean that a particular supplier has

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been chosen from the market participants and the supplier’s price for the window (cR) is fixed. Therefore, if the window is broken, then it needs to be replaced (repair/replacement cost equals the market price of the replacement), while if the window is intact then it does not need replacement (repair/replacement cost equals zero). Thus, for a particular window we have





=

= =

"

0

broken DS

c

ntact i C DS

R

R (2.10)

where cR ∈ m is the known market window replacement price and DS denotes the damage state of the window. A similar relation between damage and repair cost (or any other DV) can be obtained for every assembly. Such a relation for repair costs is routinely established by cost estimators for post-earthquake conditions once the damage to the building is known.

Therefore, once the damage state of the window is known, the repair/replacement cost can be easily found. We can use the concept of damage states for the estimation of the limit-state function (2.8). We assume that the damage state of a component is independent of the market conditions m (for example, damage state of a window does not depend on how much the window replacement cost is). It is also obvious that the damage to the window depends on the magnitude of the applied damaging factor and the ability of the window to resist this damaging factor, that is, its “strength.” In our problem, the magnitude of the damaging factor (or factors) depends on the seismic excitation q and the strength of the members and other relevant structural properties that are contained in x.

Therefore, it is reasonable to assume that the damage state of the window is a function of

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x and q: DS = gDS(q, x). The general expression for the window repair/replacement cost can be rewritten as follows

) , (

) ), , ( ( ) , , , (

R R

DS R R

c DS C

m x q g C m x q C

=

= (2.11)

It is easy to see that (2.10) gives the repair replacement cost (2.11) for the particular assembly (a window), leading to the limit-state function (2.8) trivially.

2.2.2.3 Performance-based earthquake engineering framing equation

Applying the idea of damage states to the general case, we can find the probability of DV exceeding the threshold value DVl in a way that is different from (2.9).

Consider now a case of arbitrary DV estimated for the structure under consideration and the limit-state function in the form (2.8). Suppose that the damage state is some function of the seismic excitation Q and building properties X: DS = gDS (Q, X). Suppose further that damage states are defined in such a way that DV is a function of the damage state and market conditions only

( , )

) ), , ( ( )

, , (

M DS DV

M X Q g DV M

X Q

DV _DS

=

= (2.12)

where DS is a vector of damage states of all the assemblies that affects the value of DV.

Clearly, (2.12) is generalization of (2.11) for the DV of the whole structure as opposed to the DV of a particular assembly (a window).

DV is a function of random variables and so it is a random variable itself. Thus, the probability of exceeding threshold value DVl can be found through integration of the probability density function (PDF) of DV

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∫ ( )

∞

=

>

DVL

DV

L f v dv

DV DV

P( ) (2.13)

where fDV (v) is the PDF of DV. Using (2.12), it is possible to find the PDF of DV as a marginal PDF of the joint PDF of DV and the damage measure DS

(

v ds

)

dds f

v

f_DV

∫

^∞ _DV_DS

∞

−

= ,

)

( _, (2.14)

where DS is a vector of the damage states of all the damageable components. Actually, DS is a discrete random variable whose positive values range over all the combinations of the damage states of all components. Therefore, we can count all the possible combinations and number them in some order. If there exist N different values of DS (N different combinations of damage states), then (2.14) can be rewritten as

( )

∑

=

= ^N

i

i DS DV

DV v f v ds

f

1

, ,

)

( (2.15)

For the i-th value of the damage state vector DS, we can find the joint PDF as a product of the joint probability mass function of DS and the conditional PDF of DV given DS = dsi

(

_i

)

_DV_DS

(

_i

) ( )

_DS _i

DS

DV v ds f v ds p ds

f _, , = _| | (2.16)

Consider the conditional PDF: f_DV_|_DS

(

v|ds_i

)

. It can be found by differentiating the corresponding conditional cumulative distribution function CDF: F_DV_|_DS

(

v|ds_i

)

. This CDF can be evaluated in the following way

(

_i

) (

_i

)

DS

DV v ds P DV v DS ds

F _| | = ≤ | =

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Substituting (2.12), and assuming independence of damage state and market conditions, we obtain

( ) ( )

( )

(

_v_i

)

ⁱ

i i

DS DV

M P

v ds M DV P

ds DS v ds M DV P

ds DS v DS M DV P ds v F

,

|

) , (

| ) , (

|

Ω

∈

=

≤

=

≤

=

≤

=

where Ω_v,_i is the domain in the space of market conditions variables M, which satisfies the following properties: M∈Ω_v_,_i if and only if the inequality DV(M,ds_i)≤v holds.

Thus, assuming that the joint PDF of the market conditions variables is known, the conditional CDF is found as

(

^v ^ds

)

^f

( )

^m ^d^m

F

i v

i M DS

DV

∫

Ω

=

,

| |

and the conditional PDF is

( )

f

( )

m dm

dv ds d

v f

i v

M i

DS

DV

∫

Ω

=

,

| | (2.17)

Therefore once the function DV(M,DS) is defined, the conditional PDF

(

_i

)

DS

DV v ds

f _| | can be evaluated. Substituting (2.16) and (2.15) into (2.13), the probability of DV exceeding a threshold value DVl can be written as

( ) ( )

∞

∫ ∑

=

>

DVL

N

i

i DS i DS DV

L f v ds p ds dv

DV DV P

0

| |

)

( (2.18)

where f_DV_|_DS

(

v|ds_i

)

is obtained according to (2.17).

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The joint probability mass function p_DS

( )

ds_i can be found by the approach similar to one that is used for the estimation of f_DV

( )

v . Thus, the set of structural response parameters that are vital for determining DS should be chosen. This kind of idea has been used for further modification of (2.18). As a result, the analytical methods of seismic performance evaluation for civil engineering structures with respect to multiple performance objectives have been proposed (Porter 2000, Porter et al. 2001, Beck et al.

1999, Irfanoglu 2000). Also, the integral (2.18) is rewritten in the form that is proposed to be the basis of performance-based earthquake engineering (PBEE) (Cornell and Krawinkler 2000, Krawinkler 2002, Miranda and Aslani 2003) as follows

( ) { ( ) ( )

( )

^im

} ( )

^d ^edp^d

( )

^im

f im edp f

edp dm p

dm DM DV DV P DV

DV P

IM IM

EDP

i EDP DM N

i

i l

l

)

| (

|

| 1 0 0

∑∫∫

=

∞ ∞

=

>

=

> (2.19)

where damage measure (DM) is the authors’ term for DS used before, that is, a vector containing the discrete damage states of all damageable components (henceforth, we shall use DM while considering a whole structure and DS for a separate component of the structure); EDP is a vector of engineering demand parameters containing structural response characteristics such as inter-story drift ratio (IDR), peak diaphragm acceleration (PDA), etc.; IM is an intensity measure of the ground motion such as spectral acceleration (Sa ), peak ground acceleration (PGA), etc.; ^P

(

^DV > ^DV_l |^DM = ^dm_i

)

is the probability of the decision variable being greater than DVl conditioned on knowledge of the component damage states, i = 1...N, where N is the number of possible damage states for vector DM; ^pDM_|EDP

(

^dmi |^edp

)

is the conditional probability mass function that is equal to the probability that the structure suffered the damage defined by dmi given that it

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has been subjected to the EDP whose value is equal to edp; f_EDP_|_IM(edp |im) is the conditional PDF of the structural response (EDP) given that the intensity of the ground motion is im; f_IM(im) is the PDF of the seismic event intensity measure (IM) given that an earthquake has occurred. Equation (2.19) gives the probability of a decision variable being greater than some threshold value given that an earthquake has happened.

Figure 2.3 Implementation structure of the PEER PBEE framing equation.

Note that the left-hand sides of (2.9) and (2.19) are the same and so the integrals on the right-hand side must be also equal. However, the equality of two integrals can not be taken for granted. The integration over state space va